I would like to compute the following integral: ∫ − ∞ + ∞ e i a 2 − x 2 a 2 − x 2 d x , a ∈ R

Question

I would like to compute the following integral:      ∫          −      ∞              +      ∞                  e              i                              a            2                    −                      x            2                                              a        2            −              x        2                  d    x    ,    a  ∈      R

Belen Bentley · Accepted Answer

The answer is exactly       π          H      0              (        1        )              (    a    )   for   a  &amp;gt;  0, using the Hankel functions. This can be obtained using integral representations of       J    0    (  a  ) and       Y    0    (  a  ), with your evaluation continued:                              ∫                      −            ∞                    ∞                                      e                          i                                                a                  2                                −                                  x                  2                                                                                        a              2                        −                          x              2                                              d        x                            =        2                                                                              ∫                  0                  a                                                                      e                                          i                                                                        a                          2                                                −                                                  x                          2                                                                                                                                                a                      2                                        −                                          x                      2                                                                                      d                x                            ⏟                                            x            =            a                          1              −                              t                2                                                    −        2        i                                                                              ∫                  a                  ∞                                                                      e                                          −                                                                        x                          2                                                −                                                  a                          2                                                                                                                                                x                      2                                        −                                          a                      2                                                                                      d                x                            ⏟                                            x            =            a                          1              +                              t                2                                                                                      =        2                  ∫          0          1                                                    e                              i                a                t                                                  d            t                                1            −                          t              2                                      −        2        i                  ∫          0          ∞                                                    e                              −                a                t                                                  d            t                                1            +                          t              2                                                                        =        π                              (                                    J          0                (        a        )        +        i                  Y          0                (        a        )                              )                          =        π                  H          0                      (            1            )                          (        a        )        .

I would like to compute the following integral: <msubsup> ∫ <mrow class

Answered question

Answer & Explanation

New Questions in Calculus 1